Triple Tan

Geometry Level 5

\[\large \left[ \sqrt{ \tan \dfrac {\alpha}{2} \tan \dfrac {\beta}{2} + 8} + \sqrt{ \tan \dfrac {\beta}{2} \tan \dfrac {\gamma}{2} + 8} + \sqrt{ \tan \dfrac {\gamma}{2} \tan \dfrac {\alpha}{2} + 8} \right]^2 \]

Triangle \(ABC\) has angles \( \alpha, \beta, \gamma\). What is the maximum value of the expression above.

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